Critical behaviour of random walks

TL;DR

This paper investigates the critical behavior of a two-dimensional lattice model with interacting particles and directionality, revealing phase transitions and their dependence on system parameters through numerical analysis.

Contribution

It introduces a novel lattice model with particle interaction and directionality, analyzing its phase structure and transition order via numerical simulations.

Findings

Identified a phase transition in particle density behavior.

Discovered a first-order transition at high directionality.

Computed phase diagram and critical point dependence on volume.

Abstract

We have numerically studied the trapping problem in a two-dimensional lattice where particles are continuously generated. We have introduced interaction between particles and directionality of their movement. This model presents a critical behaviour with a rich phase structure similar to spin systems. We interpret a change in the asymptotic density of particles as a phase transition. For high directionality the change is abrupt, possibly of first order. For small directionality the phase transition is of higher order. We have computed the phase diagram, the volume dependence of the critical point, and the relaxation time of the system in the large volume limit.